Self-normalized deviation inequalities with application to $t$-statistics

TL;DR

This paper derives optimal tail probability bounds for self-normalized sums of independent symmetric variables and applies these results to Student's t-statistics, enhancing understanding of their deviation behavior.

Contribution

It provides the sharpest possible bounds under Bernstein inequality assumptions for self-normalized deviations, with an application to t-statistics.

Findings

Established optimal tail bounds for self-normalized sums.

Applied bounds to analyze Student's t-statistics.

Demonstrated the bounds' tightness under Bernstein assumptions.

Abstract

Let $(\xi_i)_{i=1,...,n}$ be a sequence of independent and symmetric random variables. We consider the upper bounds on tail probabilities of self-normalized deviations $$ \mathbf{P} \Big( \max_{1\leq k \leq n} \sum_{i=1}^{k} |\xi_i|\big/ \big(\sum_{i=1}^{n} |\xi_i|^\beta \big)^{1/\beta} \geq x \Big) $$ for $x>0$ and $\beta >1.$ Our bound is the best that can be obtained from the Bernstein inequality under the present assumption. An application to Student's $t$-statistics is also given.